Question

A 16 lb weight stretches a spring 6 inches in equilibrium. It is attached to a damping mechanism with constant c. Find all values of c such that the free vibration of the weight has infinitely many oscillations.

Answer 1

To allow free vibration with infinitely many oscillations, the damping constant c must be lower than the critical damping coefficient, which is dependent on the mass and the spring constant of the system.

Step-by-step explanation:

The question is about the damping constant c in an oscillatory system, specifically a mass-spring-damper system, that will allow the mass to have infinitely many oscillations without coming to a stop. This type of motion is related to the concept of simple harmonic motion (SHM) and damped harmonic motion. The condition for infinitely many oscillations (or perpetual oscillation without coming to rest) is that the system should be critically damped or underdamped, so the value of c needs to be less than the critical damping coefficient, which can be calculated from the mass and the spring constant.

Answer 2

`-8<c<8`

Step-by-step explanation:

Given data:

Weight W = 16 lb

length l = 6/12 = 0.5 ft

hence, spring constant k = W/l = 16/0.5 = 32 lb/ft

The equation of motion of spring is

`mx''+cx+kx=0\\0.5x''+cx'+32x=0`

the auxiliary equation can be written as

`0.5r^2+cr+32=0\\r^2+2cr+64=0`

The discriminate of equation is

`D =(2c)^2+-4(1)(64)\\=4(c^2-64)`

To get the value of the damping constant,

`D<0\\4(c^2-64)<0\\c>-8, c<8\\-8<c<8`